Hi,
as an wage-earner the only option you have is what you have already found out...that you need to have been member of an A-kasse for at least 1 year and also have worked 1 year during this membership period.
The requirement about work is actually an income-requirement. It says that must have earned at least DKK 228,348 (in 2018) during the last 3 years (in total) for full-time insured.
You can read more about the requirements for being eligible for Unemployment benefits here:
https://www.a-kasser.dk/benefits/
It is not possible to pay membership fee for previous period. You can only start paying from the time you apply for membership.
Best regards,
Anders, www.a-kasser.dk